Digital Signal Processing Reference
In-Depth Information
where the leading 2
factor is for later convenience. The resulting object
is called a
pulse train
, similarly to what we built for the case of periodic se-
quences in
˜
π
N
. Using the pulse train and given any 2
π
-periodic function
l
g
r
,
y
i
d
.
,
©
,
L
s
f
(
ω
)
, the reconstruction formula (4.32) becomes
σ
+
2
π
1
2
˜
(
θ
)
˜
(
ω
)=
δ
(
θ
−
φ
)
δ
(
ω
−
φ
)
φ
f
,
f
d
(4.34)
π
σ
for any
.
Now that we have the delta notation in place, we are ready to start. First
of all, we show the formal orthogonality of the basis functions
σ
∈
e
j
ω
n
{
}
ω
∈
.
We can write
π
1
2
˜
e
j
ω
n
d
e
j
ω
0
n
δ
(
ω
−
ω
)
ω
=
(4.35)
0
π
−
π
The left-hand side of this equation has the exact form of the DTFT recon-
struction formula (4.14); hence we have found the fundamental relationship
e
j
ω
0
n
DTFT
˜
←→
δ
(
ω
−
ω
)
(4.36)
0
Now, the DTFT of a complex exponential
e
j
σ
n
is, in our change of basis in-
terpretation, simply the inner product
e
j
ω
n
,
e
j
σ
n
〈
〉
; because of (4.36) we can
therefore express this as
e
j
ω
n
,
e
j
σ
n
=
˜
δ
(
ω
−
σ
)
(4.37)
which is formally equivalent to the orthogonality relation in (4.5).
We now recall for the last time that the delta notation subsumes a limit-
ing operation: the DTFT pair (4.36) should be interpreted as shorthand for
the limit of the partial sums
k
e
−j
ω
n
s
k
(
ω
)=
n
=
−
k
(where we have chosen
ω
0
=
0forthesakeofexample). Figure4.13plots
|
s
k
(
ω
)
|
for increasing values of
k
(we show only the
[
−
π
,
π
]
interval, although
of course the functions are 2
π
-periodic). The family of functions
s
k
(
ω
)
is
exactly equivalent to the family of functions
r
k
we saw in (4.29); they too
become increasingly narrowwhile keeping a constant area (which turns out
to be 2
(
t
)
˜
π
). That is why we can simply state that
s
k
(
ω
)
→
δ
(
ω
)
.
